Question

if $$ A=\left|\begin{array}{ccc} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{array}\right| $$ then \(\mathrm{A}^{3}=\) (a) 0 (b) \(\mathrm{A}^{\mathrm{T}}\) (c) I (d) \(\mathrm{A}^{-1}\)

Short answer

The matrix \(A^3 = A\), which implies that \(A^2\) is the inverse of \(A\), i.e., \(A^2 = A^{-1}\). Therefore, the correct answer is option (d).

Step-by-step solution

Find A^2

First, we need to calculate A^2. To do this, we will multiply matrix A with itself: $$ A^2 = A \times A $$

Perform the matrix multiplication for A^2

To perform the matrix multiplication for A^2, follow the normal matrix multiplication procedure: $$ A^2=\left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{array}\right]\times\left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{array}\right] $$ $$ A^2= \left[\begin{array}{ccc} 1(1) + (-1)(2) + 1(1) & 1(-1) + (-1)(-1) + 1(0) & 1(1) + (-1)(0) + 1(0) \\ 2(1) + (-1)(2) + 0(1) & 2(-1) + (-1)(-1) + 0(0) & 2(1) + (-1)(0) + 0(0) \\ 1(1) + 0(2) + 0(1) & 1(-1) + 0(-1) + 0(0) & 1(1) + 0(0) + 0(0) \end{array}\right] $$ $$ A^2=\left[\begin{array}{ccc} 0 & 1 & 1 \\ 0 & 1 & 2 \\ 1 & -1 & 1 \end{array}\right] $$

Find A^3

Now that we have found A^2, we can find A^3 by multiplying A^2 by A: $$ A^3 = A^2 \times A $$

Perform the matrix multiplication for A^3

To perform the matrix multiplication, follow the normal matrix multiplication procedure: $$ A^3=\left[\begin{array}{ccc} 0 & 1 & 1 \\ 0 & 1 & 2 \\ 1 & -1 & 1 \end{array}\right]\times\left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{array}\right] $$ $$ A^3= \left[\begin{array}{ccc} 0(1) + 1(2) + 1(1) & 0(-1) + 1(-1) + 1(0) & 0(1) + 1(0) + 1(0) \\ 0(1) + 1(2) + 2(1) & 0(-1) + 1(-1) + 2(0) & 0(1) + 1(0) + 2(0) \\ 1(1) + (-1)(2) + 1(1) & 1(-1) + (-1)(-1) + 1(0) & 1(1) + (-1)(0) + 1(0) \end{array}\right] $$ $$ A^3=\left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{array}\right] $$ Now, we can compare the obtained matrix A^3 with the given options: (a) 0: The matrix A^3 is not a zero matrix. (b) \(A^T\): The transpose of A is: $$ A^T = \left[\begin{matrix} 1 & 2 & 1 \\ -1 & -1 & 0 \\ 1 & 0 & 0 \end{matrix}\right] $$ which is not equal to A^3. (c) I (identity matrix): The identity matrix is $$ I = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ A^3 is not equal to the identity matrix. (d) \(A^{-1}\) (inverse of A): It can be observed that A^3 = A, which implies that A^2 is the inverse of A (i.e., \(A^2 = A^{-1}\)). Therefore, the correct answer is option (d).

Key concepts

Determinants

The concept of a determinant is crucial in understanding many matrix operations, including whether or not a matrix is invertible. For a square matrix, the determinant is a scalar value that provides insight into certain properties of the matrix. Specifically, if a matrix has a determinant of 0, it is deemed singular, meaning it does not have an inverse. Calculating the determinant for a 3x3 matrix involves using the formula:

• For matrix \(A\): \[ \text{det}(A) = a(ei \text{ - } fh) - b(di \text{ - } fg) + c(dh \text{ - } eg) \]
• Where \(a, b, c, d, e, f, g, h, i\) represent the elements of the matrix.

Understanding determinants can help determine solvability in systems of linear equations and provide insights into transformations represented by matrices.

Matrix Inverse

The inverse of a matrix, denoted as \( A^{-1} \), plays a significant role in solving systems of linear equations, among other applications. A matrix must be square and have a non-zero determinant to possess an inverse. When a matrix is multiplied by its inverse, it results in the identity matrix:

• \( A \times A^{-1} = I \)

To find an inverse for a 3x3 matrix, the method involves:

• Calculating the determinant.
• Finding the matrix of minors, then the cofactor matrix.
• Taking the transpose of the cofactor matrix.
• Finally, dividing by the determinant.

Knowing how to compute an inverse is extremely useful in matrix algebra, particularly for solving equations.

Identity Matrix

An identity matrix is a pivotal concept as it acts as the multiplicative identity in matrix multiplication. For a 3x3 identity matrix, each element along the diagonal is 1, and all other elements are 0:

• \[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

The identity matrix fulfills the property:

• For any matrix \( A \), \( A \times I = A \text{ and } I \times A = A \)

This matrix is an important tool in matrix operations because it does not change any matrix it multiplies with, thus maintaining all original properties of matrix \( A \).

Matrix Transpose

A matrix transpose, denoted \( A^T \), is obtained by flipping a matrix over its diagonal. This operation switches the matrix's row and column indices. For example, if matrix \( A \) is:

• \[ A = \begin{bmatrix} 1 & -1 & 1 \ 2 & -1 & 0 \ 1 & 0 & 0 \end{bmatrix} \]

Then, its transpose \( A^T \) is:

• \[ A^T = \begin{bmatrix} 1 & 2 & 1 \ -1 & -1 & 0 \ 1 & 0 & 0 \end{bmatrix} \]

The transpose is used in various contexts, such as simplifying matrix equations or computing the inverse more efficiently. It maintains the original matrix's properties, such as determinant and rank, but presents these in a different orientation.